Observations of quiescent filaments show very fine structures which suggests that they can be composed of small-scale threads or fibrils. Two-dimensional, high-resolution observations point out that individual fibrils or groups of fibrils may oscillate independently with their own periods. In this paper, we study the fast magnetohydrodynamic modes of oscillation of multifibril Cartesian systems to represent the oscillations of the fibril structure of a real prominence. In the case of a system made of equal fibrils, our results show that the only non-leaky mode is the symmetric one, which means that all the fibrils oscillate in spatial phase with the same frequency. On the other hand, in a system made of non-equal fibrils, i.e. with different Alfvén speeds, the results show that the amplitudes of oscillation are higher in the denser fibrils, that the frequency of oscillation of the only non-leaky mode is slightly smaller than that of the dominant fibril considered alone, and that all the fibrils also oscillate in phase.